Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ Version

Zhu, Lingxue; Wu, Haijun

doi:10.1137/120874643

Cited by 98 publications

(118 citation statements)

References 52 publications

(173 reference statements)

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“…For example, with an appropriate choice of coefficients, low-order compact finite-difference discretizations can effectively reduce the dispersion error [35,58,80]. Other instances of such approaches are the generalized finite element method (GFEM) [4] and continuous interior penalty finite element method (CIP-FEM) [107,111], the interpolated optimized finite-difference method (IOFD) [93,94], Galerkin methods with hp refinement [70,72,73], among many others. These methods successfully reduce the pollution error; however, they require either a more restrictive condition on the mesh size or the degree of the polynomial approximation to be ω dependent, resulting on a large increase in the size and interconnectivity of the associated linear systems as the frequency increases.…”

Section: Related Workmentioning

confidence: 99%

“…2 Adaptive methods, on the other hand, aim to leverage à priori knowledge of the solution of the Helmholtz equation, such as its known oscillatory behavior. In practice, adaptive methods have mostly focused on adaptivity to the medium, such as polynomial Galerkin methods with hp refinement [3,70,73,96,107,111], specially optimized finite differences [23,45,92,93,102] and finite elements [4,99], enriched finite elements [30][31][32][33], plane wave methods [5,21,42,43,46,69,74], generalized plane wave methods [54,55], locally corrected finite elements [17,38,82], and discretizations with specially chosen basis functions [7,8,76], among many others. They have been especially successful on reducing the pollution effect by accurately capturing the dispersion relation.…”

Section: Introductionmentioning

confidence: 99%

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Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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We present a ray-based finite element method for the high-frequency Helmholtz equation in smooth media, whose basis is learned adaptively from the medium and source. The method requires a fixed number of grid points per wavelength to represent the wave field; moreover, it achieves an asymptotic convergence rate of O(ωwhere ω is the frequency parameter in the Helmholtz equation. The local basis is motivated by the geometric optics ansatz and is composed of polynomials modulated by plane waves propagating in a few dominant ray directions. The ray directions are learned by processing a low-frequency wave field that probes the medium with the same source. Once the local ray directions are extracted, they are incorporated into the local basis to solve the high-frequency Helmholtz equation. This process can be continued to further improve the approximations for both local ray directions and high-frequency wave fields iteratively. Finally, a fast solver is developed for solving the resulting linear system with an empirical complexity O(ω d ) up to a poly-logarithmic factor. Numerical examples in 2D are presented to corroborate the claims.

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Section: Related Workmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

et al. 2017

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“…Compared with the results on linear CIP-FEM in two and three dimensions (see [31,33,17]), the above estimate (1.2) for the case of real penalty parameters is unique in two ways. First, the mesh condition kh 1 is weaker than the requirement that k 3 h 2 is bounded, the latter being the condition required in general dimensions.…”

Section: Introductionmentioning

confidence: 64%

“…Since the phase difference of the CIP-FE solution can be reduced by tuning the penalty parameter, so can its pollution error. Recently, the authors [31,33] showed for the CIP-FEM and FEM in higher dimensions that the pollution errors in H 1 -norm are of the same order as the phase difference of the FE solution. In the higher dimensional case, although the phase difference of the CIP-FE solution can still be reduced by tuning the penalty parameter, no theoretical result says that the reduced phase difference can also control the pollution error.…”

Section: Stability and Preasymptotic Error Estimates For The Cip-femmentioning

confidence: 99%

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Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

Burman

Zhu

2016

Numer. Methods Partial Differential Eq.

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This paper addresses the properties of Continuous Interior Penalty (CIP) finite element solutions for the Helmholtz equation. The h-version of the CIP finite element method with piecewise linear approximation is applied to a one-dimensional model problem. We first show discrete well posedness and convergence results, using the imaginary part of the stabilization operator, for the complex Helmholtz equation. Then we consider a method with real valued penalty parameter and prove an error estimate of the discrete solution in the H 1 -norm, as the sum of best approximation error plus a pollution term that is the order of the phase difference. It is proved that the pollution effect can be eliminated by selecting the penalty parameter appropriately. As a result of this analysis, thorough and rigorous understanding of the error behavior throughout the range of convergence is gained. Numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior. In particular, we give numerical evidence that the optimal penalty parameter obtained in one-dimensional case also works very well for the CIP-FEM on two-dimensional Cartesian grids.

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For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

Lafontaine

Spence

Wunsch

2020

Comm Pure Appl Math

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It is well‐known that when the geometry and/or coefficients allow stable trapped rays, the outgoing solution operator of the Helmholtz equation grows exponentially through a sequence of real frequencies tending to infinity. In this paper we show that, even in the presence of the strongest possible trapping, if a set of frequencies of arbitrarily small measure is excluded, the Helmholtz solution operator grows at most polynomially as the frequency tends to infinity. One significant application of this result is in the convergence analysis of several numerical methods for solving the Helmholtz equation at high frequency that are based on a polynomial‐growth assumption on the solution operator (e.g. hp‐finite elements, hp‐boundary elements, and certain multiscale methods). The result of this paper shows that this assumption holds, even in the presence of the strongest possible trapping, for most frequencies. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

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Preasymptotic Error Analysis of CIP-FEM and FEM for Helmholtz Equation with High Wave Number. Part II: $hp$ Version

Cited by 98 publications

References 52 publications

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Learning dominant wave directions for plane wave methods for high-frequency Helmholtz equations

Linear continuous interior penalty finite element method for Helmholtz equation With High Wave Number: One-Dimensional Analysis

For Most Frequencies, Strong Trapping Has a Weak Effect in Frequency‐Domain Scattering

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